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Alexander Brudnyi - One of the best experts on this subject based on the ideXlab platform.
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holomorphic banach vector bundles on the Maximal Ideal space of h and the operator corona problem of sz nagy
Advances in Mathematics, 2013Co-Authors: Alexander BrudnyiAbstract:Abstract We establish triviality of some holomorphic Banach vector bundles on the Maximal Ideal space M ( H ∞ ) of the Banach algebra H ∞ of bounded holomorphic functions on the unit disc D ⊂ C with pointwise multiplication and supremum norm. We apply the result to the study of the Sz.-Nagy operator corona problem.
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algebras of fibrewise bounded holomorphic functions on coverings of complex manifolds ii elements of function theory
arXiv: Complex Variables, 2012Co-Authors: Alexander Brudnyi, D KinzebulatovAbstract:We establish basic results of complex function theory within certain algebras of holomorphic functions on coverings of Stein manifolds (such as algebras of Bohr's holomorphic almost periodic functions on tube domains or algebras of all fibrewise bounded holomorphic functions arising, e.g., in the corona problem for H^\infty). In particular, in this context we obtain results on holomorphic extension from complex submanifolds, properties of divisors, corona type theorems, holomorphic analogues of the Peter-Weyl approximation theorem, Hartogs type theorems, characterizations of uniqueness sets, etc. Our proofs are based on analogues of Cartan theorems A and B for coherent type sheaves on Maximal Ideal spaces of these algebras proved in Part I.
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sufficient conditions for the projective freeness of banach algebras
Journal of Functional Analysis, 2009Co-Authors: Alexander Brudnyi, Amol SasaneAbstract:Let R be a unital semi-simple commutative complex Banach algebra, and let M(R) denote its Maximal Ideal space, equipped with the Gelfand topology. Sufficient topological conditions are given on M(R) for R to be a projective free ring, that is, a ring in which every finitely generated projective R-module is free. Several examples are included, notably the Hardy algebra H∞(X) of bounded holomorphic functions on a Riemann surface of finite type, and also some algebras of stable transfer functions arising in control theory.
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grauert and lax halmos type theorems and extension of matrices with entries in h
Journal of Functional Analysis, 2004Co-Authors: Alexander BrudnyiAbstract:Abstract In the paper we prove an extension theorem for matrices with entries in H ∞ ( U ) for U a Riemann surface of a special type. One of the main components of the proof is a Grauert-type theorem for “holomorphic” vector bundles defined on Maximal Ideal spaces of certain Banach algebras.
Brett D Wick - One of the best experts on this subject based on the ideXlab platform.
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the corona theorem for the drury arveson hardy space and other holomorphic besov sobolev spaces on the unit ball in c n
Analysis & PDE, 2011Co-Authors: şerban Costea, Eric T Sawyer, Brett D WickAbstract:We prove that the multiplier algebra of the Drury-Arveson Hardy space H-n(2) on the unit ball in C-n has no corona in its Maximal Ideal space, thus generalizing the corona theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space B-p(sigma) has the "baby corona property" for all sigma >= 0 and 1 < p < infinity. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.
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the corona theorem for the drury arveson hardy space and other holomorphic besov sobolev spaces on the unit ball in mathbb c n
arXiv: Complex Variables, 2008Co-Authors: şerban Costea, Eric T Sawyer, Brett D WickAbstract:We prove that the multiplier algebra of the Drury-Arveson Hardy space $H_{n}^{2}$ on the unit ball in $\mathbb{C}^{n}$ has no corona in its Maximal Ideal space, thus generalizing the famous Corona Theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space $B_{p}^{\sigma}$ has the "baby corona property" for all $\sigma \geq 0$ and $1
Escassut Alain - One of the best experts on this subject based on the ideXlab platform.
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A survey and new results on Banach algebras of ultrametric continuous functions
HAL CCSD, 2020Co-Authors: Chicourrat Monique, Diarra Bertin, Escassut AlainAbstract:International audienceLet IK be an ultrametric complete valued field and IE be an ultrametric space. We examine some Banach algebras S of bounded continuous functions from IE to IK with the use of ultrafilters, particularly the relation of stickness. We recall and deepen results obtained in a previous paper by N. Maïnetti and the third author concerning the whole algebra A of all bounded continuous functions from IE to IK. We show that every Maximal Ideal of finite codimension of A is of codimension 1. Moreover, that property holds for every algebra S, provided IK is perfect. If S admits the uniform norm on IE as its spectral norm, then every Maximal Ideal is the kernel of only one multiplicative semi-norm, the Shilov boundary is equal to the whole multiplicative spectrum and the Banaschewski compactifiaction of IE is homeomorphic to the multiplicative spectrum of S
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Survey on the Kakutani problem in p-adic analysis II
'Academy of Sciences and Arts of Bosnia and Herzegovina', 2020Co-Authors: Escassut AlainAbstract:International audienceLet IK be a complete ultrametric algebraically closed field and let A be the Banach IKalgebra of bounded analytic functions in the "open" unit disk D of IK provided with the Gauss norm. Let M ult(A, .) be the set of continuous multiplicative semi-norms of A provided with the topology of pointwise convergence, let M ultm(A, .) be the subset of the φ ∈ M ult(A, .) whose kernel is a Maximal Ideal and let M ult1(A, .) be the subset of the φ ∈ M ult(A, .) whose kernel is a Maximal Ideal of the form (x−a)A with a ∈ D. By analogy with the Archimedean context, one usually calls ultrametric Corona problem, or ultrametric Kakutani problem the question whether M ult1(A, .) is dense in M ultm(A, .). In a previous paper, we have recalled the characterization of a large set of continuous multiplicative semi-norms and why the multbijectivity of the algebra A would solve the Corona problem. Here we are going to prove that multbijectivity in the general case which will prove that M ult1(A, .) is dense in M ultm(A, .), beginning by the case when IK is spherically complete and generalizing next
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Finite codimensional Maximal Ideals in subalgebras of ultrametric uniformly continuous functions
Belgian Mathematical Society, 2019Co-Authors: Chicourrat Monique, Diarra Bertin, Escassut AlainAbstract:International audienceLet E be a complete ultrametric space, let K be a perfect complete ultra-metric field and let A be a Banach K-algebra which is either a full K-subalgebra of the algebra of continuous functions from E to K owning all characteristic functions of clopens of E, or a full K-subalgebra of the algebra of uniformly continuous functions from E to K owning all characteristic functions of uniformly open subsets of E. We prove that all Maximal Ideals of finite codimension of A are of codimension 1. Introduction: Let E be a complete metric space provided with an ultrametric distance δ, let K be a perfect complete ultrametric field and let S be a full K-subalgebra of the K-algebra of continuous (resp. uniformly continuous) functions complete with respect to an ultrametric norm. that makes it a Banach K-algebra [3]. In [2], [4], [5], [6] we studied several examples of Banach K-algebras of functions and showed that for each example, each Maximal Ideal is defined by ultrafilters [1], [7], [8] and that each Maximal Ideal of finite codimension is of codimension 1: that holds for continuous functions [4] and for all examples of functions we examine in [2], [5], [6]. Thus, we can ask whether this comes from a more general property of Banach IK-algebras of functions, what we will prove here. Here we must assume that the ground field K is perfect, which makes that hypothesis necessary in all theorems
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Survey on the Kakutani problem in p-adic analysis I
'Academy of Sciences and Arts of Bosnia and Herzegovina', 2019Co-Authors: Escassut AlainAbstract:International audienceLet IK be a complete ultrametric algebraically closed field and let A be the Banach IK-algebra of bounded analytic functions in the "open" unit disk D of IK provided with the Gauss norm. Let M ult(A, .) be the set of continuous multiplicative semi-norms of A provided with the topology of pointwise convergence, let M ultm(A, .) be the subset of the φ ∈ M ult(A, .) whose kernel is a Maximal Ideal and let M ult1(A, .) be the subset of the φ ∈ M ult(A, .) whose kernel is a Maximal Ideal of the form (x − a)A with a ∈ D. By analogy with the Archimedean context, one usually calls ultrametric Corona problem, or ultrametric Kakutani problem the question whether M ult1(A, .) is dense in M ultm(A, .). In order to recall the study of this problem that was made in several successive steps, here we first recall how to characterize the various continuous multiplicative semi-norms of A, with particularly the nice construction of certain multiplicative semi-norms of A whose kernell is neither a null Ideal nor a Maximal Ideal, due to J. Araujo. Here we prove that multbijectivity implies density. The problem of multbijectivity will be described in a further paper
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Spectrum of ultrametric Banach algebras of strictly differentiable functions
'American Mathematical Society (AMS)', 2018Co-Authors: Escassut Alain, Maïnetti NicolasAbstract:International audienceLet IK be an ultrametric complete field and let E be an open subset of IK of strictly positive codiameter. Let D(E) be the Banach IK-algebra of bounded strictly differentiable functions from E to IK, a notion whose definition is detailed. It is shown that all elements of D(E) have a derivative that is continuous in E. Given a positive number r > 0, all functions that are bounded and are analytic in all open disks of diameter r are strictly differen-tiable. Maximal Ideals and continuous multiplicative semi-norms on D(E) are studied by recalling the relation of contiguity on ultrafilters: an equivalence relation. So, the Maximal spectrum of D(E) is in bijection with the set of equivalence classes with respect to contiguity. Every prime Ideal of D(E) is included in a unique Maximal Ideal and every prime closed Ideal of D(E) is a Maximal Ideal, hence every continuous multiplicative semi-norm on D(E) has a kernel that is a Maximal Ideal. If IK is locally compact, every Maximal Ideal of D(E) is of codimension 1. Every Maximal Ideal of D(E) is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on E. Consequently , the set of continuous multiplicative semi-norms defined by points of E is dense in the whole set of all continuous multiplicative semi-norms. The Shilov boundary of D(E) is equal to the whole set of continuous multiplicative semi-norms. Many results are similar to those concerning algebras of uniformly continuous functions but some specific proofs are required. Introduction and preliminaries
şerban Costea - One of the best experts on this subject based on the ideXlab platform.
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the corona theorem for the drury arveson hardy space and other holomorphic besov sobolev spaces on the unit ball in c n
Analysis & PDE, 2011Co-Authors: şerban Costea, Eric T Sawyer, Brett D WickAbstract:We prove that the multiplier algebra of the Drury-Arveson Hardy space H-n(2) on the unit ball in C-n has no corona in its Maximal Ideal space, thus generalizing the corona theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space B-p(sigma) has the "baby corona property" for all sigma >= 0 and 1 < p < infinity. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.
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the corona theorem for the drury arveson hardy space and other holomorphic besov sobolev spaces on the unit ball in mathbb c n
arXiv: Complex Variables, 2008Co-Authors: şerban Costea, Eric T Sawyer, Brett D WickAbstract:We prove that the multiplier algebra of the Drury-Arveson Hardy space $H_{n}^{2}$ on the unit ball in $\mathbb{C}^{n}$ has no corona in its Maximal Ideal space, thus generalizing the famous Corona Theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space $B_{p}^{\sigma}$ has the "baby corona property" for all $\sigma \geq 0$ and $1
Zhou Xiangeng - One of the best experts on this subject based on the ideXlab platform.
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On I-quotient mappings and I-cs'-networks under a Maximal Ideal
'Universitat Politecnica de Valencia', 2020Co-Authors: Zhou XiangengAbstract:[EN] Let I be an Ideal on N and f : X → Y be a mapping. f is said to be an I-quotient mapping provided f−1(U) is I-open in X, then U is I-open in Y . P is called an I-cs′-network of X if whenever {xn}n∈N is a sequence I-converging to a point x ∈ U with U open in X, then there is P ∈ P and some n0 ∈ N such that {x, xn0} ⊆ P ⊆ U. In this paper, we introduce the concepts of I-quotient mappings and I-cs′-networks, and study some characterizations of I-quotient mappings and I-cs′- networks, especially J -quotient mappings and J -cs′-networks under a Maximal Ideal J of N. With those concepts, we obtain that if X is an J -FU space with a point-countable J -cs′-network, then X is a meta-Lindelöf space.Zhou, X. (2020). On I-quotient mappings and I-cs'-networks under a Maximal Ideal. Applied General Topology. 21(2):235-246. https://doi.org/10.4995/agt.2020.12967OJS23524621
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On I-quotient mappings and I-cs'-networks under a Maximal Ideal
'Universitat Politecnica de Valencia', 2020Co-Authors: Zhou XiangengAbstract:[EN] Let I be an Ideal on N and f : X → Y be a mapping. f is said to be an I-quotient mapping provided f−1(U) is I-open in X, then U is I-open in Y . P is called an I-cs′-network of X if whenever {xn}n∈N is a sequence I-converging to a point x ∈ U with U open in X, then there is P ∈ P and some n0 ∈ N such that {x, xn0} ⊆ P ⊆ U. In this paper, we introduce the concepts of I-quotient mappings and I-cs′-networks, and study some characterizations of I-quotient mappings and I-cs′- networks, especially J -quotient mappings and J -cs′-networks under a Maximal Ideal J of N. With those concepts, we obtain that if X is an J -FU space with a point-countable J -cs′-network, then X is a meta-Lindelöf space.Zhou, X. (2020). On I-quotient mappings and I-cs'-networks under a Maximal Ideal. Applied General Topology. 21(2):235-246. https://doi.org/10.4995/agt.2020.12967OJS235246212J. R. Boone and F. Siwiec, Sequentially quotient mappings, Czech. Math. J. 26 (1976), 174-182.L. X. Cheng, G. C. Lin, Y. Y. Lan and H. Liu, Measure theory of statistical convergence, Sci. China Ser. A 51 (2008), 2285-2303. https://doi.org/10.1007/s11425-008-0017-zL. X. Cheng, G. C. Lin and H. H. Shi, On real-valued measures of statistical type and their applications to statistical convergence, Math. Comput. Modelling 50 (2009), 116-122. https://doi.org/10.1016/j.mcm.2009.04.004P. Das, Some further results on Ideal convergence in topological spaces, Topol. Appl. 159 (2012), 2621-2626. https://doi.org/10.1016/j.topol.2012.04.007P. Das and S. Ghosal, When I-Cauchy nets in complete uniform spaces are I-convergent, Topol. Appl. 158 (2011), 1529-1533. https://doi.org/10.1016/j.topol.2011.05.006P. Das, Lj.D.R. Kocinac and D. Chandra, Some remarks on open covers and selection principles using Ideals, Topol. Appl. 202 (2016), 183-193. https://doi.org/10.1016/j.topol.2016.01.003G. Di Maio and Lj. D. R. Kocinac, Statistical convergence in topology, Topol. Appl. 156 (2008), 28-45. https://doi.org/10.1016/j.topol.2008.01.015R. Engelking, General Topology (revised and completed edition), Heldermann Verlag, Berlin, 1989.H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244. https://doi.org/10.4064/cm-2-3-4-241-244L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, Princeton, 1960. https://doi.org/10.1007/978-1-4615-7819-2P. Kostyrko, T. Salát and W. Wilczynski, I-convergence, Real Anal. Exch. 26 (2000/2001), 669-686. https://doi.org/10.2307/44154069B. K. Lahiri and P. Das, I and I*-convergence in topological spaces, Math. Bohemica 130, no. 2 (2005), 153-160.S. Lin, Point-countable covers and sequence-covering mappings, Science Press, Beijing, 2015 (in Chinese).S. Lin and Z.Q. Yun, Generalized metric spaces and mapping, Atlantis Studies in Mathematics 6, Atlantis Press, Paris, 2016. https://doi.org/10.2991/978-94-6239-216-8S. K. Pal, N. Adhikary and U. Samanta, On Ideal sequence covering maps, Appl. Gen. Topol. 20, no. 2 (2019), 363-377. https://doi.org/10.4995/agt.2019.11238H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73-74. https://doi.org/10.4064/cm-2-2-98-108Z. Tang and F. Lin, Statistical versions of sequential and Fréchet-Urysohn spaces, Adv. Math. (China) 44 (2015), 945-954.X. G. Zhou and M. Zhang, More about the kernel convergence and the Ideal convergence, Acta Math. Sinica, English Series 29 (2013), 2367-2372.X. G. Zhou and L. liu, On I-covering mappings and 1-I-covering mappings, J. Math. Res. Appl. (China) 40, no. 1 (2020) 47-56.X. G. Zhou, L. Liu and S. Lin, On topological spaces defined by I-convergence, Bull. Iran. Math. Soc. 46 (2020), 675-692. https://doi.org/10.1007/s41980-019-00284-